A relation between the multiplicity of the second eigenvalue of a graph Laplacian, Courant's nodal line theorem and the substantial dimension of tight polyhedral surfaces

Abstract

A relation between the multiplicity m of the second eigenvalue λ2 of a Laplacian ona graph G, tight mappings of G and a discrete analogue of Courant’s nodal line theorem is discussed.For a certain class of graphs, it is shown that the m-dimensional eigenspace of λ2 is tight and thusdefines a tight mapping of G into an m-dimensional Euclidean space. The tightness of the mappingis shown to set Colin de Verdi´ere’s upper bound on the maximal λ2-multiplicity, m ≤ chr(γ(G)) − 1,where chr(γ(G)) is the chromatic number and γ(G) is the genus of G.